Align Eye rig to Head rig
First apply the Mirror modifier. Then remove the eyes and the head from all vertex group (in the Properties editor > Object Data > Vertex Groups, click on the down arrow menu on the right of the Vertex Groups list and choose Remove From All Groups). Now reassign each eye to the bone that is supposed to control it, and assign the head mesh to its bone. There was a bone somewhere affecting the head, it made it deform so the eyes seemed to pop out.